A spectral characterization of nonlinear normal modes

Giuseppe Cirillo (1) , Alexandre Mauroy (1) , Ludovic Renson (2) , Gaëtan Kerschen (3) , Rodolphe Sepulchre (4)

1 Université de Liège
2 Department of Engineering Mathematics
3 Space Structures and Systems Lab
4 Control Group

This paper explores the relationship that exists between nonlinear normal modes (NNMs) defined as invariant manifolds in phase space and the spectral expansion of the Koopman operator. Specifically, we demonstrate that NNMs correspond to zero level sets of specific eigenfunctions of the Koopman operator. Thanks to this direct connection, a new, global parametrization of the invariant manifolds is established. Unlike the classical parametrization using a pair of state-space variables, this parametrization remains valid whenever the invariant manifold undergoes folding, which extends the computation of NNMs to regimes of greater energy. The proposed ideas are illustrated using a two-degree-of-freedom system with cubic nonlinearity.

Mots clés: Parametrization; Invariant manifolds; Nonlinear normal modes; Koopman operator; Spectral; Characterization

Domaines: Mathématiques [math]; Sciences de l'ingénieur [physics]

Giuseppe Cirillo, Alexandre Mauroy, Ludovic Renson, Gaëtan Kerschen, Rodolphe Sepulchre. A spectral characterization of nonlinear normal modes. Journal of Sound and Vibration, 2016, 377, pp.284-301. ⟨10.1016/j.jsv.2016.05.016⟩. ⟨hal-01336227⟩